
 Prediction: at time \(t\) , an approximate function \(\hat{F}({\mathbf {s}};t+1)\)  of the true yet unobserved function \(F({\mathbf {s}};t+1)\)  is formed, using only information available at time \(t\) .
Then, using this approximated cost, we derive an estimate \({\mathbf {s}}_{t+1|t}^\star \) , of how the topology will be at time \(t+1\) , using only the information up to time \(t\) . This estimate is found by solving:
\({\mathbf {s}}_{t+1\vert t}^\star := \operatornamewithlimits{argmin}_{\mathbf {s}}\hat{F}({\mathbf {s}};t+1).\) 

To avoid solving (REF ) for each \(t\) , we find an estimate \(\hat{{\mathbf {s}}}_{t+1\vert t}\)  by applying \(P\)  iterations of a problem-specific descent operator \(\hat{{\mathcal {T}}}\)  (e.g., gradient descent, proximal gradient) for which \({\mathbf {s}}_{t+1\vert t}^\star = \hat{{\mathcal {T}}} {\mathbf {s}}_{t+1\vert t}^\star \) , i.e., \({\mathbf {s}}_{t+1\vert t}^\star \)  is a fixed point of \(\hat{{\mathcal {T}}}\) . See Appendix  for possible instances of \(\hat{{\mathcal {T}}}\) .
In other words, problem (REF ) is solved recursively as:
\(\hat{{\mathbf {s}}}^{p+1}= \hat{{\mathcal {T}}}\hat{{\mathbf {s}}}^p, \quad p=0,1, \ldots , P-1\) 

with \(\hat{\bf s}^0=\hat{\bf s}_t\) .
Once \(P\)  steps are performed, the predicted topology is set to \(\hat{{\mathbf {s}}}_{t+1 \mid t}= \hat{{\mathbf {s}}}^{P}\) , which approximates the solution of (REF ) and, in turn, will be close to \({\mathbf {s}}_{t+1}^{\star }\)  at time \(t+1\) .
For our framework, we consider a Taylor-expansion based prediction to approximate the first term of \(F(\cdot ; t+1)\) , i.e., \(f(\cdot ; t+1)\)  [cf. (REF )], leading to the following quadratic function:
\(\nonumber \hat{f}({\mathbf {s}}; t+1)&= \frac{1}{2} {\mathbf {s}}^\top \nabla _{{\mathbf {s}}{\mathbf {s}}} f\left(\hat{{\mathbf {s}}}_{t} ; t\right) {\mathbf {s}}+\big [\nabla _{{\mathbf {s}}} f\left(\hat{{\mathbf {s}}}_{t} ; t\right)+ \\&+ h \nabla _{t {\mathbf {s}}} f\left(\hat{{\mathbf {s}}}_{t} ; t\right)-\nabla _{{\mathbf {s}}{\mathbf {s}}} f\left(\hat{{\mathbf {s}}}_{t} ; t\right) \hat{{\mathbf {s}}}_{t}\big ]^\top {\mathbf {s}}\) 

where \(\nabla _{{\mathbf {s}}{\mathbf {s}}} f (\cdot ) \in \mathbb {R}^{N \times N}\)  is the Hessian matrix of \(f(\cdot )\)  with respect to \({\mathbf {s}}\)  and \(\nabla _{t {\mathbf {s}}} f(\cdot ) \in \mathbb {R}^{N}\)  is the partial derivative of the gradient of \(f(\cdot )\)  w.r.t. time \(t\) .
To approximate the second term of \(F(\cdot ; t+1)\) , i.e., \(g(\cdot ; t+1)\)  [cf. (REF )], we use a one step-back prediction, i.e., \({\hat{g}({\mathbf {s}}; t+1)= g({\mathbf {s}}; t)}\) . This implies that \(\hat{g}(\cdot )\)  does not depend on \(t\) , which in turn makes the constraint set and the regularization term independent of time, an assumption usually met in state-of-the-art topology identification [1]}. Henceforth, we will omit this time dependency.

Correction: at time \(t+1\)  the new data \({\mathbf {x}}_{t+1}\)  and hence the cost function \(F({\mathbf {s}};t+1)\)  becomes available. Thus, we correct the prediction \(\hat{{\mathbf {s}}}_{t+1\vert t}\)  by solving the correction problem:
\({\mathbf {s}}_{t+1}^\star := \operatornamewithlimits{argmin}_{\mathbf {s}}F({\mathbf {s}};t+1).\) 

Also in this case, we solve (REF ) with iterative methods to obtain an approximate solution \(\hat{{\mathbf {s}}}_{t+1}\)  by applying \(C\)  iterations of an operator \({\mathcal {T}}\) . In other words, the correction problem (REF ) is addressed through the recursion:
\(\hat{{\mathbf {s}}}^{c+1}= {\mathcal {T}}\hat{{\mathbf {s}}}^c, \quad c=0,1, \ldots , C-1\) 

with \(\hat{\bf s}^0=\hat{\bf s}_{t+1|t}\) . Once the \(C\)  steps are performed, the correction graph \(\hat{{\mathbf {s}}}_{t+1}\)  is set to \(\hat{{\mathbf {s}}}_{t+1}= \hat{{\mathbf {s}}}^{C}\) , which will approximate the solution \({\mathbf {s}}_{t+1}^{\star }\)  of (REF ).

